Phonon mode frequencies acoustic

In parallel with these experimental studies of PL for SQD suspensions, the role of the observed linewidth broadening has been examined. In particular, the linewidth broadening due to acoustic-phonon-assisted transitions is expected [1] to contribute to satellite lines in PL spectra that are downshifted by the acoustic phonon energies. Within the elastic continuum approach [6], the phonon mode frequencies sensitive to the boundary conditions at the SQD surface were calculated. The lowest-order spherical acoustic mode frequency for CdS for different matrix materials differ by as much as a factor of three for a given SQD radius. 

Fig. 40. (a) Temperature dependence of the longitudinal acoustic-phonon frequencies of Smo Y jsS in the [111] direction for four different values of the wavevector q (see Mook et al. 1981). (b) Temperature dependence of the bulk modulus Cg of Sm Y 25S measured by Bril-iouin scattering. Cg continues to soften upon cooling below 200 K, uniike the behavior of the phonon mode frequencies for qaO.l (flg. 40a). (c) Temperature dependence of the charge relaxation rate derived from the experimental data in figs. 40a and 40b (open circles) and calculated from theory (Schmidt and Miiller-Hartmann 1985) (solid line). The theoretical curve has been matched at 300 K to the experimental value. 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

When the displacements of the nuclei are considered in terms of the phonon modes of the crystal, the reorganization energy can be expressed in terms of the relative shifts of the equilibrium positions of acoustic vibrations of acoustic vibrations. Using the Debye model and assuming that the friction is independent of frequency, the reorganization energy is proportional to the friction coefficient, Et = AI2>7Ty]0 2DQ (see Section 2.3). 

By comparing the resonance frequency Eq.(ll) and the phonon vibration frequency Eq.(12), we see that they are almost the same, 0.3 0.4 x 1014 s 1. This affirms the possibility of a spin-paired covalent-bonded electronic charge transfer. For vibrations in a linear crystal there are certainly low frequency acoustic vibrations in addition to the high frequency anti-symmetric vibrations which correspond to optical modes. Thus, there are other possibilities for refinement. In spite of the crudeness of the model, this sample calculation also gives a reasonable transition temperature, TR-B of 145 °K, as well as a reasonable cooperative electronic resonance and phonon vibration effect, to v. Consequently, it is shown that the possible existence of a COVALON conduction as suggested here is reasonable and lays a foundation for completing the story of superconductivity as described in the following. 

Here, M is the reduced mass for the optic oscillation in the cell, or the mass of crystal cell for the acoustic phonons mk the mode frequency and N the number of cells in the macro-crystal. It is easy to see that the sum in Eq. (11) converges for all types of the displacements 8Rt due to the rapid decrease in 8Rt with the increasing distance from the center Rt. Therefore, the lattice particles located near the center only give the real contribution to the sum. The number N is very large, so the displacement 8qg for each mode is very small. Then, one may take into account the first few terms only in the expansion of the final phonon wave function on displacement 8q ... 

The quite another temperature dependence of the rate constant at helium temperatures is resulted in the case when the principal contribution to dispersion a in formula (25a) gives the acoustic phonons. Their frequencies lie in the interval [0, lud], where tuD is Debye s frequency. Even if hin0 kT, it exists always in the range of such low frequencies that haxkT. It is these phonons that give the contribution depending on the temperature in the dispersion a [15], One assumes that the displacements of the equilibrium positions of phonon modes Sqs do not depend on frequency. Then, the calculation of the rate constant gives at low temperatures, hcou>kT,... 

The optical spectral region consists of internal vibrations (discussed in Section 1.13) and lattice vibrations (external). The fundamental modes of vibration that show infrared and/or Raman activities are located in the center Brillouin zone where k = 0, and for a diatomic linear lattice, are the longwave limit. The lattice (external) modes are weak in energy and are found at lower frequencies (far infrared region). These modes are further classified as translations and rotations (or librations), and occur in ionic or molecular crystals. Acoustical and optical modes are often termed phonon modes because they involve wave motions in a crystal lattice chain (as demonstrated in Fig. l-38b) that are quantized in energy. 

The condition for observation is that the phonon coherence length is larger than the layer thickness. Low frequency acoustic modes fulfill this condition because they are an in-phase motion of a large number of atoms and are not strongly influenced by the disorder-instead reflecting the average bulk elastic properties of the materials. 

The dispersion relationships of lattice waves may be simply described within the first Brillouin zone of the crystal. When all unit cells are in phase, the wavelength of the lattice vibration tends to infinity and k approaches zero. Such zero-phonon modes are present at the center of the Brillouin zone. The variation in phonon frequency as reciprocal k) space is traversed is what is meant by dispersion, and each set of vibrational modes related by dispersion is a branch. For each unit cell, three modes correspond to translation of all the atoms in the same direction. A lattice wave resulting from such displacements is similar to propagation of a sound wave hence these are acoustic branches (Fig. 2.28). The remaining 3N-3 branches involve relative displacements of atoms within each cell and are known as optical branches, since only vibrations of this type may interact with light. 

Wurtzite ZnO structure with four atoms in the unit cell has a total of 12 phonon modes (one longitudinal acoustic (LA), two transverse acoustic (TA), three longitudinal optical (LO), and six transverse optical (TO) branches). The optical phonons at the r point of the Brillouin zone in their irreducible representation belong to Ai and Ei branches that are both Raman and infrared active, the two nonpolar 2 branches are only Raman active, and the Bi branches are inactive (silent modes). Furthermore, the Ai and Ei modes are each spht into LO and TO components with different frequencies. For the Ai and Ei mode lattice vibrations, the atoms move parallel and perpendicular to the c-axis, respectively. On the other hand, 2 modes are due to the vibration of only the Zn sublattice ( 2-low) or O sublattice ( 2-high). The expected Raman peaks for bulk ZnO are at 101 cm ( 2-low), 380 cm (Ai-TO), 407 cm ( i-TO), 437 cm ( 2-high), and 583 cm ( j-LO). 

Consider now the case where the energy spacing 21 is very small. Such cases are encountered in the study of relaxation between spin levels of atomic ions embedded in crystal environments, so called spin-lattice relaxation. The spin level degeneracy is lifted by the local crystal field and relaxation between the split levels, caused by coupling to crystal acoustical phonons, can be monitored. The relaxation as obtained from (12.47) and (12.48) is very slow because the density of phonon modes at the small frequency (U21 is small (recall that... 

Fig. 44. Schematic representation of the influence of different charge relaxation rates of IV rare-earth ions on the frequencies of the optical phonon modes (e.g., (dj and at ) and on the q=0 longitudinal acoustic-phonon modes, represented by the bulk modulus Cg (see text). For the stable n - and (n -1- l) -valent rare-earth compounds we show generalized reference lines. Four typical cases are shown with the representative samples given at the bottom.

Phonon bands occur in the SBZ, similarly to the surface states discussed in Sect. 5.2.3. When the frequency of a surface mode corresponds to a gap in the bulk spectrum, the mode is localized at the surface and is called a surface phonon. If degeneracy with bulk modes exists, one speaks of surface resonances. Surface phonon modes are labeled Sj ( / = 1, 2, 3,...), and surface resonances by Rj when strong mixing with bulk modes is present, the phonon is labeled MSj. The lowest mode that is desired from the (bulk) acoustic band is often called the Rayleigh mode, after Lord Rayleigh, who first predicted (in 1887) the existence of surface modes at lower frequencies than in the bulk. 

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